2 - Thermodynamics in a Vacuum

Thermodynamics in a Vacuum

Once a silica nanoparticle is securely caught in the optical trap, a new challenge arises: we cannot actually see it. At 100 nm in diameter, the particle is smaller than the wavelength of the trapping light itself (1550 nm), putting it well beneath the diffraction limit of standard optical microscopy.

Instead of looking at the particle, we “listen” to it. We collect the laser light that scatters off the particle and interfere it with the unscattered beam on a high-speed photodetector. This gives us a voltage signal that corresponds directly to the particle’s position.

Brownian Motion and the Time Trace

If you look at the raw voltage output on an oscilloscope, it looks like pure static. This is Brownian motion. Even though the trap is holding the particle in place, the particle is constantly being bombarded by the air molecules inside the vacuum chamber. Because the trap acts like a 3D spring (a harmonic oscillator), the particle rapidly bounces back and forth around the center point, driven by thermal energy and damped by air resistance.

The Power Spectral Density (PSD)

To make sense of the noise, we use a Fast Fourier Transform (FFT) to convert the time-domain signal into the frequency domain, creating a Power Spectral Density (PSD) plot. Suddenly, the random noise resolves into three beautiful, distinct Lorentzian peaks. Each peak represents the particle’s oscillation in one spatial dimension:

1. The Z-Axis (Low Frequency): The weakest trapping axis, parallel to the laser beam.
2. The X & Y Axes (High Frequency): The transverse axes. They are slightly split in frequency because the laser polarization (controlled by the waveplates) breaks the symmetry of the trap.

Interactive Calibration

Use the dashboard below to explore how the physical environment affects the particle’s motion.

• The Vacuum Pump: Lower the pressure to remove air resistance. Watch the time trace amplitude increase and the PSD peaks become incredibly sharp (a high Q-factor).
• Laser Power: Adjust the trap stiffness. Higher power pushes the resonant frequencies higher up the spectrum.
• Waveplates: Tweak the HWP and QWP to change the polarization ellipticity, which directly controls the frequency split between the X and Y axes.

By carefully fitting theoretical curves to these three PSD peaks, experimentalists can perfectly calibrate the trap, determining the exact mass of the particle and the local temperature to incredibly high precision. This calibration is the mandatory first step before attempting to cool the particle into the quantum regime.

The Optical Potential Well

Before we look at the data, we must understand the mechanics of the trap itself. To a silica nanoparticle, the focused laser beam acts exactly like a 3D microscopic spring. In physics, we call this a Harmonic Oscillator. When the particle is sitting perfectly in the center of the laser focus, it feels no net force. But if it drifts away from the center (due to thermal energy from the environment), the electromagnetic gradient pushes it back. This restorative force is described by Hooke’s Law: where is the trap stiffness (determined by the laser power) and is the displacement from the center.

Because of this force, the particle lives inside an optical potential well. It is constantly rolling back and forth at the bottom of a parabolic “bowl” of light.

Interactive: The Harmonic Oscillator

In the simulation below, you are looking at a 1D slice of this potential well. Try increasing the Trap Stiffness (turning up the laser power). Notice how the “bowl” becomes steeper, and the particle oscillates at a much higher frequency. Then, turn up the Thermal Kicks to see how the surrounding gas molecules bump the particle around inside the well, preventing it from ever sitting perfectly still.